3.7 \(\int \frac {\cot ^{-1}(a x)}{x} \, dx\)

Optimal. Leaf size=37 \[ \frac {1}{2} i \text {Li}_2\left (\frac {i}{a x}\right )-\frac {1}{2} i \text {Li}_2\left (-\frac {i}{a x}\right ) \]

[Out]

-1/2*I*polylog(2,-I/a/x)+1/2*I*polylog(2,I/a/x)

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Rubi [A]  time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4849, 2391} \[ \frac {1}{2} i \text {PolyLog}\left (2,\frac {i}{a x}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-\frac {i}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/x,x]

[Out]

(-I/2)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4849

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I/(c*
x)]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {align*} \int \frac {\cot ^{-1}(a x)}{x} \, dx &=\frac {1}{2} i \int \frac {\log \left (1-\frac {i}{a x}\right )}{x} \, dx-\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{a x}\right )}{x} \, dx\\ &=-\frac {1}{2} i \text {Li}_2\left (-\frac {i}{a x}\right )+\frac {1}{2} i \text {Li}_2\left (\frac {i}{a x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 37, normalized size = 1.00 \[ \frac {1}{2} i \text {Li}_2\left (\frac {i}{a x}\right )-\frac {1}{2} i \text {Li}_2\left (-\frac {i}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]/x,x]

[Out]

(-1/2*I)*PolyLog[2, (-I)/(a*x)] + (I/2)*PolyLog[2, I/(a*x)]

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fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x,x, algorithm="fricas")

[Out]

integral(arccot(a*x)/x, x)

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giac [A]  time = 0.13, size = 38, normalized size = 1.03 \[ -\frac {1}{2} \, {\left (\frac {x^{2} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {x}{a^{2}} + \frac {\arctan \left (\frac {1}{a x}\right )}{a^{3}}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x,x, algorithm="giac")

[Out]

-1/2*(x^2*arctan(1/(a*x))/a + x/a^2 + arctan(1/(a*x))/a^3)*a^2

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maple [B]  time = 0.06, size = 63, normalized size = 1.70 \[ \ln \left (a x \right ) \mathrm {arccot}\left (a x \right )-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {i \dilog \left (i a x +1\right )}{2}+\frac {i \dilog \left (-i a x +1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)/x,x)

[Out]

ln(a*x)*arccot(a*x)-1/2*I*ln(a*x)*ln(1+I*a*x)+1/2*I*ln(a*x)*ln(1-I*a*x)-1/2*I*dilog(1+I*a*x)+1/2*I*dilog(1-I*a
*x)

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maxima [B]  time = 0.46, size = 56, normalized size = 1.51 \[ \frac {1}{4} \, \pi \log \left (a^{2} x^{2} + 1\right ) - \arctan \left (a x\right ) \log \left (a x\right ) + \operatorname {arccot}\left (a x\right ) \log \relax (x) + \arctan \left (a x\right ) \log \relax (x) + \frac {1}{2} i \, {\rm Li}_2\left (i \, a x + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-i \, a x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x,x, algorithm="maxima")

[Out]

1/4*pi*log(a^2*x^2 + 1) - arctan(a*x)*log(a*x) + arccot(a*x)*log(x) + arctan(a*x)*log(x) + 1/2*I*dilog(I*a*x +
1) - 1/2*I*dilog(-I*a*x + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acot}\left (a\,x\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)/x,x)

[Out]

int(acot(a*x)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}{\left (a x \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/x,x)

[Out]

Integral(acot(a*x)/x, x)

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